A theorem on roots of unity and a combinatorial principle
arXiv:1409.5822
Abstract
Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the non-negativity of certain alternating sums is equivalent to the set system being a filter. As an application we determine all discrete Fourier pairs of -matrices. This technical result is an essential step in the classification of -matrices of quantum groups.
We have proven the more general combinatorial statement and made some other minor improvements